Contributions in optimal sampled-data control theory with state constraintsand nonsmooth data

This dissertation is concerned with first-order necessary optimality conditions in the form of a Pontryagin maximum principle (in short, PMP) for optimal sampled-data control problems with free sampling times, running inequality state constraints and nonsmooth Mayer cost functions.Chapter 1 is devoted to notations and basic framework needed to describe the optimal sampled-data control problems to be encountered in the manuscript. In Chapter 2, considering that the sampling times can be freely chosen, we obtain an additional necessary optimality condition in the PMP called the Hamiltonian continuity condition. Recall that the Hamiltonian function, which describes the evolution of the Hamiltonian taking values of the optimal trajectory and of theoptimal sampled-data control, is in general discontinuous when the sampling times are fixed. Our result proves that the continuity of the Hamiltonian function is recovered in the case of optimal sampled-data controls with optimal sampling times. Finally we implement a shooting method based on the Hamiltonian continuity condition in order to numerically determine the optimal sampling times in two linear-quadratic examples.In Chapter 3, we obtain a PMP for optimal sampled-data control problems with running inequality state constraints. In particular we obtain that the adjoint vectors are solutions to Cauchy-Stieltjes problems defined by Borel measures associated to functions of bounded variation. Moreover, we find that, under certain general hypotheses, any admissible trajectory (associated to a sampled-data control) necessarily bounces on the runningine quality state constraints. Taking advantage of this bouncing trajectory phenomen on, we are able to use thePMP to implement an indirect numerical method which we use to numerically solve some simple examples of optimal sampled-data control problems with running inequality state constraints. In Chapter 4, we obtain a PMP for optimal sampled-data control problems with nonsmooth Mayer cost functions. Our proof directly follows from the tools of nonsmooth analysis and does not involve any regularization technique. We determine the existence of a selection in the subdifferential of the nonsmooth Mayer cost function by establishing a more general result asserting the existence a universal separating vector for a given compact convex set. From the application of this result, which is called universal separating vector theorem, we obtain a PMP for optimal sampled-data control problems with nonsmooth Mayer cost functions where the transversality conditon on the adjoint vector is given by an inclusion in the subdifferential of the nonsmooth Mayer cost function.To obtain the optimality conditions in the form of a PMP, we use different techniques of perturbations of theoptimal control. In order to handle the state constraints, we penalize the distance to them in a corresponding cost functional and then apply the Ekeland variational principle. In particular, we invoke some results on renorming Banach spaces in order to ensure the regularity of distance functions in the infinite-dimensional context. Finally we use standard notions in nonsmooth analysis such as the Clarke generalized directional derivative and theClarke subdifferential to study optimal sampled-data control problems with nonsmooth Mayer cost functions.